Download Joint distribution of the multivariate Ornstein-Uhlenbeck

The 22nd Annual Meeting in Mathematics (AMM 2017)
Department of Mathematics, Faculty of Science
Chiang Mai University, Chiang Mai, Thailand
Joint distribution of the multivariate
Ornstein-Uhlenbeck process
Pat Vatiwutiponga,b,‡ and Nattakorn Phewcheana,b,†
a
Department of Mathematics, Faculty of Science
Mahidol University, Bangkok 10400, Thailand
b
Centre of Excellence in Mathematics
CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand
Abstract
In this paper, we derive a joint probability density function of the multivariate OrnsteinUhlenbeck process by solving the Fokker-Planck equation. This approach allows us to know
its distribution without solving analytically. We found that, at each time, the process
have a multivariate normal distribution. Explicit formula for each parameters are obtained.
Moreover, when time t tends to infinity, with some specific condition, this process will tend
to be stationary.
Keywords: multivariate Ornstein-Uhlenbeck process, multivariate normal distribution, FokkerPlanck equation, n-dimensional Fourier transform.
2010 MSC: 93E03.
1
Introduction
Stochastic models have been used to forecast a fluctuation value over time for many years.
Although deterministic model is easier to use, stochastic models become more popular, because
of the fact that the noise terms are considered. The Ornstein-Uhlenbeck process is one of the
most well-known stochastic process that use in mathematical finance. Oldrich Vasicek (1977)
applied this process to model the instantaneous interest rate over times (see [1]). This models
were also used in another research areas. For example the Hookean spring whose dynamics is
highly over-damped with friction coefficient in physics (see [2]). The membrane potential of
a neuron which is perturbed by electrical impulses from the surrounding network in biology
(see [3]). Analytic solution, function of mean, variance and covariance of it overtime t were
derived.
The problem is when we want to model many values together, univariate models force us
to consider it independently, which is not a realistic assumption. In some situation, we might
do like that, but it is certainty not work for the case that each values are related. That is
why the idea of multivariate models were arisen. Multivariate Ornstein-Uhlenbeck process is
a generalized process of the Ornstein-Uhlenbeck process. It can be describe intuitively as a
†
Corresponding author.
Speaker.
E-mail address: [email protected] (P. Vatiwutipong), [email protected] (N. Phewchean).
‡
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massive random particle under the influence of several frictions. There are many studies and
applications of it such as [4], [5], [6] and [7].
However, most of these past studies focus on solving for its solution (both analytically and
numerically) and study on its parameters estimation. They consider it as a solution of stochastic
differential equation, so many tools in stochastic calculus such as multivariate Ito’s Lemma which
require a lots of background knowledge in stochastic calculus were needed. Alternative, this
study consider its Fokker-Plank equation which is an another representation of this process. We
use an n-dimensional Fourier transform, characteristic method and some background knowledge
in matrix calculus to derive the joint probability density function and its parameters.
2
Preliminaries
In this section, we will introduce some useful definitions and results.
Firstly, we recall the multivariate Fokker-Plank equation, which is one of the most important
equation in stochastic theory. For more details, we suggest Chapter 3-5 of [8].
For a multivariate Ito process X(t) defined by the stochastic differential equation
dX(t) = µ(X(t), t)dt + σ(X(t), t)dW (t),
(2.1)
where µ(X(t), t) is an n-dimensional vector, σ(X(t), t) is an n × m matrix and W (t) is an mdimensional standard Wiener process. The probability density function p(x, t) of X(t) satisfies
the Fokker-Planck equation
∂
∂
∂ ∂
p(x(t), t) = −
· [µ(x(t), t)p(x(t), t)] +
: [D(x(t), t)p(x(t), t)],
∂t
∂x
∂x ∂x
(2.2)
where D(x(t), t) = σ(X(t), t)σ T (X(t), t) (for a complete derivation, see [9]). This equation
are also known as the Kolmogorov forward equation.
In general, there are no way to solve this equation analytically. For our case, n-dimensional
Fourier transform can be applied. We recall the definition and some of its properties.
Let f : Rn → R to be continuous. The n-dimensional Fourier transform of f is the function
F(f ) : Rn → R defined to be
∫
F(f )(u) =
f (x)e−i(x·u) dx,
Rn
where i is an imaginary unit.
Proposition 2.1. Let f : Rn → R such that continuously partial differentiable and lim||x||→∞ f (x) =
0. For any n × n real matrix A and n-dimensional real vector c, these following properties hold
∂
1. F( ∂x
· cf (x)) = iuT cF(f )(u),
∂
2. F( ∂x
: Af (x)) = −uT AuF(f )(u),
(
)T
)(u)
∂
3. F( ∂x
· Axf (x)) = − ∂F(f
AT u.
∂u
These properties can be derived directly by Proposition 6.8.1 of [10].
∑
Ak
For any square matrix A, we define the exponential of A, denote eA , to be ∞
k=0 k! where
A0 is defined to be the identity matrix I. Note that this series always converge so the definition
is well-defined. There are some properties of matrix exponential that we will use later on, so
we will summaries it here. For more details, see Chapter 2 of [11].
Proposition 2.2. For any square matrix A, the following properties hold
1. AeA = eA A,
T
2. (eA )T = eA ,
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3. eA is invertible with e−A as its inverse,
4.
3
∫
deAt
= AeAt , so if A is invertible, eAt dt = A−1 eAt .
dt
Main Results
Definition 3.1. An n-dimensional Ornstein-Uhlenbeck process X(t) is a multivariate stochastic
process satisfying
(
)
dX(t) = θ µ − X(t) dt + σdW (t),
(3.1)
where θ is an n × n invertible real matrix, µ is an n-dimensional real vector, σ is an n × m
positive real matrix and W (t) is an m-dimensional standard Wiener process.
Theorem 3.2. The characteristic function of n-dimensional Ornstein-Uhlenbeck process X(t)
satisfying (3.1) with the initial value X(0) = x0 is given by
[
ϕ(u) = exp iu
(
T
1 (
e−θt x0 + (I − e−θt )µ − uT
2
)
∫
t
eθ(s−t) σσ T e
θ T (s−t)
]
)
ds u .
(3.2)
0
Proof. We starting from the Fokker-Planck equation of (3.1):
[
]
[
]
∂
∂
1 ∂ ∂
∂p
=−
· θµp −
· θxp +
: Dp ,
∂t
∂x
∂x
2 ∂x ∂x
(3.3)
where D = σσ T , with initial condition
(
)
p(x) = δ x − x0 .
2
(3.4)
By taking the n-dimensional Fourier transform of equation (3.3), we get
( ∂ p̂ )T
∂ p̂
1
= −iuT θµp̂ +
θ T u − uT Dup̂,
∂t
∂u
2
(3.5)
where p̂(u, t) is the n-dimensional Fourier transform of p(x, t).
The initial condition (3.4) becomes
)
(
p̂(u0 ) = exp − iuT0 x0 .
(3.6)
Equation (3.5) is a first order partial differential equation, so we apply the method of
characteristic.
Firstly, we solve the system
du
= θ T u,
dt
with initial condition u(0) = u0 . The solution is
T
u = eθ t u0 .
Consider another characteristic equation
[
]
dp̂
1 T
T
= − iu θµ − u Du p̂.
dt
2
Substitute (3.7) in (3.8), getting
[
]
dp̂
1 T θt θT t
T θt
= − iu0 e θµ − u0 e De u0 dt.
p̂
2
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(3.7)
(3.8)
(3.9)
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Integrate both side, and get
[
p̂ = p̂0 exp −
iuT0 (eθt
1 (
− I)µ − uT0
2
∫
t
θt
e De
θT t
) ]
dt u0 .
(3.10)
0
Then, substitute p̂0 from (3.6) and u0 by inverting (3.7) into (3.10), we get
[
]
∫
1 ( t θt θT t )
p̂ = exp − iuT0 x0 − iuT0 (eθt − I)µ − uT0
e De dt u0
2
0
[
]
∫
1 T −θt ( t θt θT t ) −θT t
T −θt
T −θt θt
= exp − iu e x0 − iu e (e − I)µ − u e
e De dt e
u
2
0
[
]
∫
1 T ( t θ(s−t) T θT (s−t) )
T −θt
T
−θt
= exp − iu e x0 − iu (I − e )µ − u
e
σσ e
ds u .
2
0
(3.11)
Since the characteristic function is its Fourier transform with opposite sign in the complex
exponential, we are done.
Corollary 3.3. The n-dimentianal Ornstein-Uhlenbeck process X(t) satisfying (3.1) have a
n-dimentianal normal distribution with mean vector and covariance matrix given by
M (t) = e−θt X0 + (I − e−θt )µ,
(3.12)
and
∫
t
eθ(s−t) σσ T eθ
Σ(t) =
T (s−t)
ds
(3.13)
0
respectively. Moreover, the probability density function of X(t) is given by
(
(
)T
(
))
exp − 12 x − M (t) Σ−1 (t) x − M (t)
√
p(x, t) =
.
|2πΣ(t)|
(3.14)
Proof. Compare (3.2) with a characteristic function of multivariate normal distribution
1
ϕ(u) = exp[iuT M − uT Σu],
2
(3.15)
then we obtain the result.
As we know that the univariate Ornstein-Uhlenbeck process, with positive reverting rate θ,
is tend to be stationary when t is increasing. For multivariate case, it also have a mean-reverting
condition which is stated in the following corollary.
Corollary 3.4. The n-dimentianal Ornstein-Uhlenbeck process X(t) satisfying (3.1) will tend
to be stationary when t increase if all eigenvalues of θ are positive. This property is called mean
reversion.
Proof. Since e−θt converts to the zero matrix if all eigenvalues of θ are positive, we can conclude
from (3.12) that, with this condition, M (t) converts to µ.
For Σ(t) is difference, we cannot take t in (3.13) to infinity directly as we do for M (t). We
applied the identity vec(ABC) = (C T ⊗ A) vec(B) where ⊗ is a Kronecker product defined
in [12] and vec(A) is defined to be a column vector made of the columns of A stacked atop one
another from left to right. Then
∫
vec(Σ(t)) =
t
eθ(s−t) ⊗ eθ(s−t) ds vec(σσ T ).
(3.16)
0
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Now we use another identity eA⊗B = eA ⊕ eB where ⊕ is a Kronecker sum. Then we obtain
∫
t
vec(Σ(t)) =
eθ(s−t) ⊗ eθ(s−t) ds vec(σσ T )
0
∫
t
e(θ⊕θ)(s−t) ds vec(σσ T )
0
(
)
= (θ ⊕ θ)−1 I − e−(θ⊕θ)t vec(σσ T ).
=
(3.17)
Since all eigenvalues of θ ⊕ θ are still positive, the covariance matrix converts to constant
matrix Σ such that vec(Σ) = (θ ⊕ θ)−1 vec(σσ T ).
4
Conclusion
This paper provide an alternative method to get the probability distribution of multivariate
Ornstein-Uhlenbeck by solving its Fokker-Planck equation. This method allows us to get the
result without using tools in stochastic calculus. Moreover, the mean-reverting condition is
obtained. Although we do not know the exact value of this process at each time (since it is a
stochastic model), our results can provide its distribution, mean and variance. These datas can
be used to forecast its trend. For future studies, in order to extend our results, we may consider
the multivariate mean-reverting process in general.
Acknowledgment. This work was completed with the support of Centre of Excellence in
Mathematics
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